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Beltrami Flow
In fluid dynamics, Beltrami flows are flows in which the vorticity vector \mathbf and the velocity vector \mathbf are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881. Description Since the vorticity vector \boldsymbol and the velocity vector \mathbf are parallel to each other, we can write :\boldsymbol\times\mathbf=0, \quad \boldsymbol = \alpha(\mathbf,t) \mathbf, where \alpha(\mathbf,t) is some scalar function. One immediate consequence of Beltrami flow is that it can never be a planar or axisymmetric flow because in those flows, vorticity is always perpendicular to the velocity field. The other important consequence will be realized by looking at the incompressible vorticity equation :\frac + (\mathbf\cdot\nabla)\bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lamb Vector
In fluid dynamics, Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb.Truesdell, C. (1954). The kinematics of vorticity (Vol. 954). Bloomington: Indiana University Press. The Lamb vector is defined as :\mathbf l = \mathbf \times \boldsymbol where \mathbf is the velocity field and \boldsymbol=\nabla\times\mathbf is the vorticity field of the flow. It appears in the Navier–Stokes equations through the material derivative term, specifically via convective acceleration term, : \mathbf u\cdot \nabla\mathbf u = \frac\nabla\mathbf u^2 - \mathbf\times\boldsymbol = \frac\nabla\mathbf u^2 - \mathbf l In irrotational flows, the Lamb vector is zero, so does in Beltrami flows. The concept of Lamb vector is widely used in turbulent flows. The Lamb vector is analogous to electric field, when the Navier–Stokes equation is compared with Maxwell's equations. Properties of Lamb vector The divergence of the lamb vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Riley (professor)
Norman Riley is an Emeritus Professor of Applied Mathematics at the University of East Anglia in Norwich (UK). Biography Following High School education at Calder High School, Mytholmroyd he read Mathematics at Manchester University graduating with first class honours in 1956, followed by a PhD in 1959. Norman Riley served for one year as an Assistant Lecturer at Manchester University and then spent four years as a lecturer at Durham University before he joined the then new University of East Anglia in 1964, the year that saw the first significant intake of students to the university. Promotion to Reader in 1966 was followed by promotion to a Personal Chair in 1971. He retired in 1999. Married in 1959 he has one son and one daughter. Research contributions His research contributions in the field of fluid mechanics, over five decades, have included: unsteady flows with application to acoustic levitation and the loading on the submerged horizontal pontoons of tethered leg platf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chia-Shun Yih
Chia-Shun Yih (; July 25, 1918 – April 25, 1997) was the Stephen P. Timoshenko Distinguished University Professor Emeritus at the University of Michigan. He made many significant contributions to fluid mechanics. Yih was also a seal artist. Biography Yih was born on July 25, 1918, in Guiyang, Guizhou province of China. Yih received his junior middle school education in Zhenjiang, and entered Suzhou High School in 1934 in Suzhou, Jiangsu Province. In 1937, Yih entered the National Central University and studied civil engineering. Yih graduated in 1941 then did research at a hydrodynamics laboratory in Guanxian (or Guan County; 灌县; current Dujiangyan) of Sichuan province. Yih also worked in a bridge construction company in Guizhou. Later, Yih taught at Guizhou University. In 1945, Yih went to study at the University of Iowa in the United States, where he obtained his PhD in 1948. Yih served as a professor of the University of Michigan for most of his academic career. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hagen–Poiseuille Equation
In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846. The theoretical justification of the Poiseuille law was given by George Stokes in 1845. The assumptions of the equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. For velocities and pipe diam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function Of The Second Kind
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function Of The First Kind
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hicks Equation
In fluid dynamics, Hicks equation, sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation, is a partial differential equation that describes the distribution of stream function for axisymmetric inviscid fluid, named after William Mitchinson Hicks, who derived it first in 1898. The equation was also re-derived by Stephen Bragg and William Hawthorne in 1950 and by Robert R. Long in 1953 and by Herbert Squire in 1956. The Hicks equation without swirl was first introduced by George Gabriel Stokes in 1842. The Grad–Shafranov equation appearing in plasma physics also takes the same form as the Hicks equation. Representing (r,\theta,z) as coordinates in the sense of cylindrical coordinate system with corresponding flow velocity components denoted by (v_r,v_\theta,v_z), the stream function \psi that defines the meridional motion can be defined as :rv_r = - \frac, \quad rv_z = \frac that satisfies the continuity equation for axisymmetric flows automatically. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as "one of the most notable scientists of this (the 20th) century". Early life and education Taylor was born in St. John's Wood, London. His father, Edward Ingram Taylor, was an artist, and his mother, Margaret Boole, came from a family of mathematicians (his aunt was Alicia Boole Stott and his grandfather was George Boole). As a child he was fascinated by science after attending the Royal Institution Christmas Lectures, and performed experiments using paint rollers and sticky-tape. Taylor read mathematics and physics at Trinity College, Cambridge from 1905 to 1908. Then he obtained a scholarship to continue at Cambridge under J. J. Thomson. Career and research To students of physics, Taylor is best known for his very first paper, publishe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blasius Boundary Layer
In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow ( Falkner–Skan boundary layer), i.e. flows in which the plate is not parallel to the flow. Prandtl's boundary layer equations Using scaling arguments, Ludwig Prandtl argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate). This leads to a reduced set of equations known as the boundary layer equations. For steady incompressible flow with constant viscosity and density, these read: Mass Continuity: \dfrac+\dfrac=0 x-Momentum: u \dfrac + v \dfrac = - \dfrac \dfrac + \dfrac y-Momentum: 0= - \dfrac \dfrac Here the coordinate system is chosen wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Of Variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) Suppose a differential equation can be written in the form :\frac f(x) = g(x)h(f(x)) which we can write more simply by letting y = f(x): :\frac=g(x)h(y). As long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, so that the two variables ''x'' and ''y'' have been separated. ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Alternative notation Those who dislike Leibniz's notation may prefer to write this as :\frac \frac = g(x), but th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Drazin
Philip Gerald Drazin (25 May 1934 – 10 January 2002) was a British mathematician and a leading international expert in fluid dynamics. He completed his PhD at the University of Cambridge under G. I. Taylor in 1958. He was awarded the Smith's Prize in 1957. After leaving Cambridge, he spent two years at MIT before moving to the University of Bristol, where he stayed and became a Professor until retiring in 1999. After retiring, he lectured at the University of Oxford and the University of Bath until his death in 2002. Drazin worked on hydrodynamic stability and the transition to turbulence. His 1974 paper ''On a model of instability of a slowly-varying flow'' introduced the concept of a global mode solution to a system of partial differential equations such as the Navier-Stokes equations. He also worked on solitons. In 1998 he was awarded the Symons Gold Medal of the Royal Meteorological Society. References External links Philip Gerald Drazinat the Mathematics Geneal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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